Sum of Squares Decompositions for Structured Biquadratic Forms

Abstract

This paper studies sum-of-squares (SOS) representations for structured biquadratic forms. We prove that diagonally dominated symmetric biquadratic tensors are always SOS. For the special case of symmetric biquadratic forms, we establish necessary and sufficient conditions for positive semi-definiteness of monic symmetric biquadratic forms, characterize the geometry of the corresponding positive semi-definite (PSD) cone as a convex polyhedron, and prove that every such PSD form is SOS for any dimensions m and n. We also raise two open questions regarding SOS representations for symmetric M-biquadratic tensors and symmetric $\textrm{B}_{0}$-biquadratic tensors. Our results advance the understanding of when positive semi-definiteness implies SOS decompositions for structured biquadratic forms.

Keywords

Biquadratic forms / Sum-of-squares (SOS) / Positive semi-definiteness / M-eigenvalues / Symmetric bilinear forms / 11E25 / 12D15 / 14P10 / 15A69 / 90C23

References

[1] Calderón AP. A note on biquadratic forms. Linear Algebra Appl., 1973, 7: 175-177.

[2] Chen H, Li G, Qi L. SOS tensor decomposition: theory and applications. Commun. Math. Sci., 2016, 14: 2073-2100.

[3] Choi M-D. Positive semidefinite biquadratic forms. Linear Algebra Appl., 1975, 12: 95-100.

[4] Cui C, Qi L, Xu Y. Positive semidefinite and sum of squares biquadratic polynomials. Mathematics, 2025, 13: 2294.

[5] Ding W, Liu J, Qi L, Yan H. Elasticity M-tensors and the strong ellipticity condition. Appl. Math. Appl., 2020, 373: 124982

[6] Goel C, Kuhlmann S, Reznick B. The analogue of Hilbert’s 1888 theorem for even symmetric forms. J. Pure Appl. Algebra, 2017, 221: 1438-1448.

[7] Hilbert, D.: Über die Darstellung definiter Formen als Summe von Formenquadraten. Math. Ann. 32, 342–350 (1888)

[8] Qi L, Cui C. Biquadratic tensors: eigenvalues and structured tensors. Symmetry, 2025, 17: 1158.

[9] Qi, L., Cui, C., Chen, H., Xu, Y.: Completely positive biquadratic tensors. Appl. Math. Lett. 175, 109849 (2026)

[10] Qi, L., Cui, C., Xu, Y.: The SOS rank of biquadratic forms. arXiv:2507.16399v4 (2025)

[11] Qi, L., Dai, H.H., Han, D.: Conditions for strong ellipticity and M-eigenvalues. Front. Math. China 4, 349–364 (2009)

[12] Qi L, Luo Z. Tensor Analysis: Spectral Theory and Special Tensors, 2017. Philadelphia, SIAM.

[13] Wang, G., Sun, L., Liu, L.: M-eigenvalues-based sufficient conditions for the positive definiteness of fourth-order partially symmetric tensors. Complexity 2020, 1–8 (2020)

[14] Zhao J. Conditions of strong ellipticity and calculation of M-eigenvalues for a partially symmetric tensor. Appl. Math. Comput., 2023, 458: 128245

Funding

The Hong Kong Polytechnic University(4-ZZT8)

National Natural Science Foundation of China(12471282 and 12131004)

Graduate Research and Innovation Projects of Jiangsu Province(BK20233002)

The Hong Kong Polytechnic University

RIGHTS & PERMISSIONS

The Author(s)